Sine, Cosine and Tangent

The three main functions in trigonometry are Sine, Cosine and Tangent.

You are watching: Tangent is positive in quadrants i and iv only


They are straightforward to calculate:

Divide the length of one side of aright angled triangle by another side

... Yet we must recognize which sides!

Using this triangle (lengths are only to one decimal place):

sin(35°) = opposite / Hypotenuse = 2.8/4.9 = 0.57...

Cartesian Coordinates

Using Cartesian collaborates we note a suggest on a graph by how much along and how much up it is:

The suggest (12,5) is 12 devices along, and 5 devices up.


Four Quadrants

When we include negative values, the x and y axes division the space up right into 4 pieces:

Quadrants I, II, III and IV

(They room numbered in a counter-clockwise direction)

In Quadrant I both x and y are positive, in Quadrant II x is an adverse (y is still positive), in Quadrant III both x and y are negative, andin Quadrant IV x is confident again, and y is negative.

Like this:



Example: The point "C" (−2,−1) is 2 units along in the negativedirection, and also 1 unit under (i.e. Negative direction).

Both x and y room negative, for this reason that suggest is in "Quadrant III"

Sine, Cosine and Tangent in theFour Quadrants

Now let united state look in ~ what happens when we location a 30° triangle in every of the 4 Quadrants.

In Quadrant I every little thing is normal, and also Sine, Cosine and also Tangent room all positive:

Example: The sine, cosine and also tangent the 30°


sin(30°) = 1 / 2 = 0.5
cos(30°) = 1.732 / 2 = 0.866
tan(30°) = 1 / 1.732 = 0.577

But in Quadrant II, the x direction is negative, and also both cosine and also tangent end up being negative:

Example: The sine, cosine and also tangent the 150°


sin(150°) = 1 / 2 = 0.5
cos(150°) = −1.732 / 2 = −0.866
tan(150°) = 1 / −1.732 = −0.577

In Quadrant III, sine and also cosine room negative:

Example: The sine, cosine and tangent the 210°


sin(210°) = −1 / 2 = −0.5
cos(210°) = −1.732 / 2 = −0.866
tan(210°) = −1 / −1.732 = 0.577

Note: Tangent is positive because dividing a an adverse by a an adverse gives a positive.

In Quadrant IV, sine and tangent space negative:

Example: The sine, cosine and tangent that 330°


sin(330°) = −1 / 2 = −0.5
cos(330°) = 1.732 / 2 = 0.866
tan(330°) = −1 / 1.732 = −0.577

There is a pattern! look at when Sine Cosine and also Tangent space positive ...

All 3 of them space positivein Quadrant ISine just is positive in Quadrant IITangent only is hopeful in Quadrant IIICosine only is hopeful in Quadrant IV

This have the right to be presented even less complicated by:


Some civilization like come remember the four letters ASTC by one of these:

All Students take it ChemistryAll Students take it CalculusAll stunner Tom CatsAll Stations come CentralAdd Sugar To Coffee

You have the right to remember among these, or possibly you can make upyour own. Orjust remember ASTC.

This graph mirrors "ASTC" also.

two Values

Have a look at this graph that the Sine Function::

There space two angles (within the very first 360°) that have the very same value!

And this is additionally true for Cosine and Tangent.

The trouble is: Your calculator will only give you one of those values ...

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... However you can use this rules to uncover the various other value:

First valueSecond value
Sineθ180º − θ
Cosineθ360º − θ
Tangentθθ − 180º

And if any kind of angle is much less than 0º, then add 360º.

We can now solve equations forangles in between 0º and also 360º(using train station Sine Cosine and Tangent)

Example: fix sin θ = 0.5

We obtain the an initial solution from the calculator = sin-1(0.5) = 30º(it is in Quadrant I)

The other solution is 180º − 30º = 150º (Quadrant II)

Example: Solvetan θ= −1.3

we getthe first solution native the calculator = tan-1(−1.3) = −52.4º

This is much less than 0º, for this reason we add 360º: −52.4º + 360º = 307.6º (Quadrant IV)

The othersolution is307.6º − 180º =127.6º (Quadrant II)

Example: Solvecos θ= −0.85

us getthe very first solution indigenous the calculator = cos-1(−0.85) =148.2º (Quadrant II)

The various other solution is 360º −148.2º = 211.8º (Quadrant III)